Specialization and Nonrenewable Resources: Ricardo meets Ricardo*
by
Ujjayant Chakravorty, Department of Economics, Emory University
Darrell Krulce, QUALCOMM, Inc., San Diego
and
James Roumasset, Department of Economics, University of Hawaii at Manoa.
Abstract
The one-demand Hotelling model fails to explain the observed specialization of nonrenewable
resources. We develop a model with multiple demands and resources to show that specialization
of resources according to demand is driven by Ricardian comparative advantage while the order
of resource use over time is determined by Ricardian absolute advantage. An abundant resource
with absolute advantage in all demands must be initially employed in all demands. When each
resource has an absolute advantage in some demand, no resource may be used exclusively. The
two-by-two model is characterized. Resource and demand-specific taxes are shown to have
significant substitution effects.
*
We would like to thank Mike Caputo, Bob Chirinko, Joy Mazumdar and Paul Rubin, and especially Peter
Ireland, Editor of this journal for valuable suggestions and comments. This work was partially supported by
National Science Foundation grant SES 91-22370.
Email address for correspondence: Ujjayant Chakravorty, Department of Economics, Emory University,
Atlanta, GA 30322, phone: (404)-727-1381; fax: (404)-727-4639; email address: unc@emory.edu.
JEL Classification: D9, Q3, Q4
Keywords: Comparative advantage, Dynamic models, Energy resources, Heterogenous demand,
Hotelling theory
2
1. Introduction
The modern theory of resource economics (Hotelling, 1931; Herfindahl, 1967), which traces its roots
to Ricardo’s theory of the mine,1 has largely focused on the problem of one resource and a single
demand. However, casual empirical observation suggests that different nonrenewable resources (e.g.,
oil, coal) and grades (e.g., crude oils from Saudi Arabia and Alaska) are being extracted
simultaneously to satisfy distinct energy demands. For example, Table 1 shows the composition of
global energy supply by sector and resource for a recent year (2000). Notice that crude oil and
petroleum products are the fuels of choice in the transportation sector, coal in industry and natural gas
in industrial and residential uses. Coal, nuclear, natural gas and hydroelectric energy supply the bulk
of electricity, an intermediate good used mainly in the industrial and residential sectors. The singledemand model does not adequately explain this simultaneity and specialization of resources in
specific sectors.
Not only are multiple nonrenewable resources being extracted simultaneously, but the pattern of
resource substitution is induced by a host of economic factors (e.g., demand growth, the
discovery of new reserves, and technological change) that may vary between sectors and between
resources. A recent International Energy Agency (IEA) study of energy demand until 2020
suggests that in the OECD countries, growth in consumption in the transport sector is entirely
accounted for by oil, while in the residential, industry and electricity sectors, oil continues to lose
market share to other fuels, especially natural gas (IEA 2000). In particular, natural gas, in recent
years, has become competitive in the electricity sector because of the advent of combined cycle
gas turbine technology. In the past, electricity was generated predominantly from coal, but new
power plants today mostly use natural gas, except in countries with large indigenous reserves of
1
Chapter 3 in Ricardo (1912), first edition, 1817.
3
coal. This trend is significant enough that the IEA expects global energy supplies from natural gas
to surpass that of coal around 2010. The shift from coal to natural gas is also evident in the
industry and residential sectors. The transition from coal and oil to the cleaner natural gas and the
switch from oil to coal in industry after the OPEC oil price shocks of the seventies suggests that
correctly specified empirical models that study the evolution of long-run resource price
movements need to allow for fuel switching within and between sectors. 2
The class of models following from Hotelling and Herfindahl (1967), who extended Hotelling to
multiple grades of a resource (still with a single demand), cannot explain this apparent
specialization of resources in particular demands because in the single demand framework,
different resources are indistinguishable from different grades of the same resource. Due to this
restriction, empirical tests of resource prices based on the Hotelling theory may be mis-specified
(see e.g., Halvorsen and Smith, 1991).3 In this paper, we extend the above class of HotellingHefindahl models to a general multiple demand framework that allows for specialization of
resources and substitution across demands based on notions of absolute and comparative
advantage. We show that this extension generates an equilibrium sequence of resource extraction
2
Another reason why a multiple (rather than a single) demand model may be a more appropriate
framework for analysis is the significant difference in growth characteristics among sectors. For example,
annual average growth in global demand in the electricity (2.8%) and transport sectors (2.4%) is markedly
higher than that in other uses such as residential and industry (1.8%), according to IEA (2000).
3
In his Richard T. Ely address to the American Economic Association, Arnold Harberger (1993) expressed the
same sentiment when he suggested that resource economics is a prominent example of economic theory failing
to live up to the demands of its practitioners. The gap between theory and practice can be partially closed by
developing a theory of multiple resources that are imperfect substitutes in meeting various resource demands.
4
and energy prices over time that is quite distinct from those derived from standard Hotelling
theory.
There are several papers that have extended Hotelling to multiple resources and demands,
beginning with Herfindahl (1967) as noted earlier, who extended the Hotelling model to m
resources and a single demand. Kemp and Long (1980), Lewis (1982), and Amigues et al. (1998)
have generalized the basic m x 1 model by developing sufficient conditions under which
Herfindahl’s “least cost first” principle (the ordering of resource extraction by their unit cost)
holds and conditions under which it may not hold.
Nordhaus (1973, 1979) pioneered the extension to the m x n (m resources and n demands) case in
an applied study concerning the long-run tendency of energy prices. Chakravorty, Roumasset and
Tse (1997) generalized and applied the Nordhaus framework to examine the effect of exogenous
changes in the price of the backstop technology on fossil fuel extraction and carbon emissions
over time. These applied studies do not develop the analytics of the m x n model. Chakravorty
and Krulce (1994) provided a theoretical treatment of a 2 x 2 case (two resources and two
demands) when one resource has absolute advantage over the other. However, as will
become clear in this paper, their characterization of the 2 x 2 model is a special case.4 The
objective of the present paper is to develop the general theory for the m x n model and then to
completely characterize the 2 x 2 case.
4
They developed a 2 x 2 model in an infinite horizon framework where one resource (oil) has absolute
advantage over the other in both demands. They showed that under these specific conditions, it will always
be the case that a more expensive resource will be used for a finite time interval even though the cheaper
resource is not exhausted, violating the well known “least cost first” principle proposed by Herfindahl. The
objective of their paper was to show that this principle need not hold in a multiple demand, partial
5
The m x n framework affords an analytical distinction between resources and resource grades.
We assume that there is a constant extraction cost for each grade but that different grades of the
same resource may have different extraction costs. We follow Nordhaus in assuming that energy
demands in different sectors are independent.5 Solutions to an infinite horizon maximization
problem yield equilibrium relationships for a given resource in a given demand in terms of the
scarcity rent and cost characteristics of the resource.6
We show that patterns of resource use can be characterized by stages according to which
resources are used in what demands. We exploit Ricardian notions of absolute and comparative
advantage in characterizing dynamic patterns of resource specialization. For example, if one
equilibrium framework. However, they did not proceed to develop the full implications of the multiple
demand framework.
5
This assumption of independence is supported by the fact that seasonal shocks such as the effect of
summer driving on demand for transportation energy or a cold winter on residential energy demand tend to
be sector-specific.
6
Gaudet, Moreaux and Salant (2001) have examined a somewhat analogous general problem in which
solid wastes are transported from urban centers to spatially distributed landfills. In their model, landfill
capacity is exhaustible, and landfills are differentiated by transportation costs from each city. The landfill
model is similar to ours in that transportation costs from cities to landfills can be thought of as conversion
costs of resources to demands. However, resources are differentiable by class (oil, coal) and by grade
(different grades of oil) while landfills are homogenous except for their location. While Gaudet et al. focus
on the role of set up costs, we characterize patterns of optimal resource use according to principles of
absolute and comparative advantage of a resource. Thus their work although in a spatial urban economics
setting, may be thought of as a special case of our model in which each resource is of a different class. In
6
resource has a Ricardian absolute advantage over all other resources and is sufficiently abundant,
it will be used exclusively for all end uses in the first stage. However, when each resource has an
absolute advantage in some demand in a m x m model, a resource can never supply all demands,
however abundant it may be. A strictly inferior resource will be used in all end uses in the final
stage, regardless of its initial abundance. The two-resource two-demand case is completely
characterized under conditions in which one resource has an absolute advantage in both uses, and
when both resources have absolute advantage in some demand.
Our results suggest that absolute advantage leads to dynamic specialization while comparative
advantage results in intersectoral specialization. A resource that is abundant and has absolute
advantage in all demands may be extracted for a demand even though it does not have
comparative advantage in that use. However if each resource has absolute advantage in a given
demand, then specialization is likely to occur on the basis of comparative advantage. Ricardian
comparative advantage thus provides a basis for developing the dynamic Ricardian/Hotelling
theory of resource rents.
We show that taxes on a resource or on a sector (e.g., transportation) in the multiple demand
model may have effects that are quite different than in the standard Hotelling model, which only
predicts lower resource use over time. For example, a tax on coal may lead to both sectors
switching to oil, while a tax on the transportation sector may lead to both sectors using oil, so that
aggregate oil consumption may increase. A resource and sector-specific tax (such as a gasoline
tax) may lead to a complete switch in resource use between sectors. That is, if oil was being used
for transportation and coal for electricity ex-ante, coal would be used for transportation and oil for
later work it may be useful to develop a general model with both resource-independent transportation costs
and demand-specific conversion costs.
7
electricity after the tax. Finally, predictions on energy price paths can be obtained directly from
the sum of extraction plus conversion costs of resources. In the case of “clean” oil and “dirty”
coal, we show that sectoral energy price differentials must decline over time, in sharp contrast to
standard Hotelling theory, which predicts prices in both sectors to increase at the rate of discount.
Section 2 develops the general m x n Hotelling model. Section 3 characterizes the 2 x 2 case.
Section 4 concludes the paper.
2. The m x n Model
Consider a finite set of resources R (e.g. various grades of oil, coal, natural gas, etc.) and a finite
set of uses for these resources defined by the set U (such as electricity, heating, transportation,
etc.). The available stock of resource i Є R, assumed known, is Qi(t0) > 0 which can be extracted
at a constant unit cost of ci ≥ 0. Derived demand for energy in use j Є U is a strictly positive,
∞
bounded, continuous, strictly decreasing function of price, Dj(p) with
∫ D ( p )dp < ∞.
j
This last
0
restriction implies a finite consumer surplus and is useful in guaranteeing a solution to the
problem. Energy for the same use generated from different resources is assumed to be identical
and the differences between resources subsumed in conversion costs. For example, coal can either
be liquefied and used to produce a gasoline substitute, or car engines can be designed to run on
coal, whichever is cheaper. The conversion cost of coal for use in transportation is then derived
from the lesser of these costs. Conversion costs are resource and demand-specific such that there
is a vector mapping from each resource to the set of demands. They are denoted by vij ≥ 0, which
is the cost of converting a unit of resource i to use j. Energy losses, such as frictional, heat or
handling losses in the conversion process, are assumed to be incorporated into the cost of
8
conversion and already netted out of the resource endowments.7 The demand relationship is in
terms of “delivered” energy units. We define the net cost of supplying resource i to demand j as
wij ≡ ci + vij.
The social planner chooses the quantity of each resource supplied to each demand. We denote by
qij(t) the quantity of resource i supplied to demand j at time t. The problem is to determine the
resource allocation that maximizes the present value of net social benefit. Given a discount rate
r>0, this can be posed as the optimal control problem: choose qij(t) for i Є R and j Є U to
maximize
∞
∫e
− rt
t0
Z
[ ∑ ∫ D −j 1 ( x )dx − ∑ ∑ wij qij ( t )] dt
j∈U 0
(1)
i∈R j∈U
subject to
qij(t) ≥ 0, Qi(t) ≥ 0 for i Є R and j Є U, and
Q& i (t) = -
∑q
j ∈U
where
Z =
ij
(2)
(t) for i Є R,
(3)
∑ qij ( t ) , aggregate energy consumption in demand j at time t. The state variable Qi(t) is
i∈ R
the residual stock of resource i over time. The two terms in (1) denote the standard sum of
consumer plus producer surplus. The current value Hamiltonian for the above problem is given
by
7
In empirical applications, efficiency losses can be made explicit (see e.g. Nordhaus, 1979, Chakravorty et
al., 1997).
9
Z
H = ∑ ∫ D −j 1 ( x )dx − ∑ ∑ wij qij ( t )] dt − ∑ λi ( t )∑ qij ( t )
j∈U 0
i∈R j∈U
i∈R
j∈U
where λi(t) ≥ 0 has the standard interpretation as the scarcity rent of resource i. The solution is
defined in terms of optimal price paths as functions of time. Let the price of the resource input for
−1
demand j be p j ( t ) ≡ D j (
Q& i (t) = -
∑q
i∈R
ij
( t ) ). The necessary conditions for a solution are:
∑ q (t) for i Є R ,
(4)
ij
j ∈U
λ& ( t ) = rλi ( t ) for i ∈ R ,
(5)
pj(t) ≤ wij + λi(t) (if < then qij(t)=0) for i Є R and j Є U, and
(6)
lim e − rt λi ( t )Qi ( t ) = 0 for i Є R .
(7)
i
t →∞
It is straightforward to show that a solution to the above program exists:
Proposition 1. There exists a unique optimal solution to program (1)-(3) and the necessary
conditions (4)-(7) are also sufficient.
Proof: See Appendix 1.
Before proceeding, we prove the intuitive but useful result that all resources approach exhaustion
in the limit.
Lemma 1. lim Qi ( t ) = 0 for i Є R.
t →∞
10
Proof. Pick a Є R and suppose that λa(t0)=0. From (5), λa(t)=0 and so from (6), pj(t)≤ waj. Since
demand is positive and downward sloping, for some j Є U,
m
∞ m
i =1
t 0 i =1
0 < D j ( waj ) ≤ D j ( p j ( t )) = ∑ qij ( t ). Thus
∞
∫q
bj
∫∑q
ij
( t )dt = ∞ so there exists b Є R such that
( t )dt = ∞. From (4), Q& b ( t ) = − ∑ qbj ( t ) ≤ −qbj ( t ) and so eventually Qb(t) will become
j∈U
t0
negative which contradicts (2). Thus the supposition is false and so λa(t0) > 0. Combining (5) and
(7) yields 0 = lim e − rt λ a ( t )Qa ( t ) = lim e − rt λ a ( t 0 )e rt Qa ( t ) = λ a ( t 0 ) lim Qa ( t ) which since
t →∞
t →∞
t →∞
λa(t0) > 0 implies that lim Qa ( t ) = 0. Since a was arbitrary, then lim Qi ( t ) = 0 for i Є R . ■
t →∞
t →∞
Hotelling Scarcity Rents in the Heterogenous Demand Framework
The necessary conditions (4) through (7) can be easily interpreted. Condition (4) is just a
restatement of (3). Condition (5) is the familiar Hotelling equation which suggests that scarcity
rents rise over time at the rate of discount. Condition (6) is the basic Kuhn-Tucker condition
governing resource allocation in the multiple demand model that says that the price in any given
demand cannot exceed the net cost of any resource in that demand. This inequality implies that
the resource that is available at the lowest price (net cost plus scarcity rent) is always used for
each demand, as proved by the following proposition:
Proposition 2. The price (net cost plus scarcity rent) of a resource that is supplied for a given
demand is no more than that of any alternative resource.
Proof: Suppose that qaj(t) > 0 for some a Є R, j Є U and t Є (t0,∞). From (6) waj+ λa(t) = pj(t) ≤
wij+ λi(t) for i Є R . ■
Solving (5) produces the familiar Hotelling equation
11
λi ( t ) = λi ( t0 )e rt for i Є R
(8)
which states that the scarcity rent rises at the rate of discount. Condition (8) also implies that the
scarcity rents of all resources are ordered. Based on this ordering, we write λa < λb to mean λa(t) <
λb(t) for all t Є [t0,∞). It may also be the case that the scarcity rents of two resources are the same.
As shown by the following proposition, this must be the case if two resources ever
simultaneously supply the same demand.
Proposition 3. Two resources simultaneously supplying the same demand have the same scarcity
rent and net cost for that demand.
Proof. Let qaj(t) > 0 and qbj(t) > 0 for some a,b Є R, j Є U, and t Є I where I ⊂ ( t 0 , ∞ ) is an open
interval. From (6),
waj + λ a ( t ) = p j ( t ) = wbj + λb ( t ) for t Є I .
(9)
Differentiating, we get λ&a ( t ) = λ&b ( t ) for t Є I. From (5) and (8), λa=λb. That is, the scarcity
rents are the same. Combining with (9) yields waj = wbj. That is, the net costs are equal. ■
A corollary to the above result is:
Proposition 4. The prices of two demands that are simultaneously supplied by the same resource
must grow at the same rate.
12
Proof: Let resource i supply both demands j and k over an interval I ⊂ ( t 0 , ∞ ) .Then
p j ( t ) = wij + λi ( t ), p k ( t ) = wik + λi ( t ) which implies that p& j ( t ) = λ&i ( t ) = p& k ( t )∀t ∈ I . ■
Ricardian Absolute Advantage
In the standard Hotelling/Herfindahl model with a single demand, resource rents are ordered by
grade: the resource with the highest grade has the highest scarcity rent. With heterogenous
demand, the ordering of resource rents is more problematic since there is not necessarily an
ordering of costs among multiple resources. One resource may be cheaper for one demand and
more costly for another demand when compared to other resources. The following definitions
relate three different types of cost orderings that may occur:
Definition. Resource a Є R has an absolute advantage relative to resource b Є R in demand j Є
U if waj < wbj , some j Є U.
Definition. Resource a Є R dominates resource b Є R, if it has an absolute advantage in all
demands, i.e., if waj < wbj , all j Є U.
Definition. Resource a Є R is universally dominant if it dominates all other resources, i.e., if waj
< wij, all i Є R, j Є U.
Absolute advantage implies lower net cost relative to another resource in a single demand. A
resource that dominates another, has Ricardian absolute advantage relative to this other resource
in all demands. A resource that universally dominates has absolute advantage, i.e., is strictly
cheaper, relative to all resources and for all demands. The next results generalize the principle of
cost-ordered scarcity rents.
13
Proposition 5. Dominant resources have a higher scarcity rent.
Proof: Let waj < wbj for resources a,b Є R and all demands j Є U. Suppose that λa ≤ λb. From (6),
pj(t) ≤ waj + λa(t) < wbj + λb(t) for all t Є (t0,∞). Thus qbj(t)=0 for all t Є (t0,∞) and j Є U; resource
b is never extracted for any demand. Since this contradicts the Lemma, the supposition is false
and thus λa > λb. ■
By the above proposition, a resource that universally dominates (if one exists) will have the
highest scarcity rent, since its scarcity rent will be higher than that of all other resources. Note
that it is not possible to order resources with only absolute advantage by their scarcity rents. This
is because resource a (b) may have absolute advantage over resource b (a) in demand j (k), so
that no resource is dominant.
Resources and Grades
The multiple-demand framework allows for a clear distinction between individual resources and
grades of a single resource:
Definition. Resource a,b Є R are of the same resource class if wbj – waj = d for all j Є U and some
constant d. Furthermore, if d > 0 (d = 0, d < 0) then resource a is a higher grade (same grade,
lower grade) of the resource class than resource b.
This formal definition of resource class corresponds roughly to what is meant in common
parlance by distinguishing between resources of different types, e.g. “coal”, “oil”, “natural gas,”
etc. The difference between resources within any class is only cost – a higher-grade resource has
the same cost advantage regardless of demand. This classification is based on the economic
properties of the resource, not its chemical properties. Two resources with similar chemical
14
compositions, e.g. light vs heavy crude oil, may be in different resource classes, depending on
whether their cost advantage varies across uses. In the Herfindahl case of one demand, all
resources are in the same resource class, i.e. there is no distinction between different resources
and different grades of the same resource.
Proposition 6. Higher-grade resources have a larger scarcity rent.
Proof: If resource a Є R is a higher grade of the same resource class than resource b Є R, then by
definition, wbj – waj = d > 0 for j Є U which implies that waj < wbj for j Є U. From Proposition 5,
resource a has a larger scarcity rent than resource b. ■
That is, a higher grade of the same resource class dominates the grade to which it is being
compared, and the highest grade dominates all other grades. With only one resource class but
multiple demands, the highest grade is universally dominant. With homogenous (one) demand,
the Herfindahl principle states that resources are extracted sequentially, in order of cost. The
following two propositions generalize this principle to show that the use of resources within each
demand is always in order of absolute advantage and that resources are extracted by decreasing
grade.
Proposition 7. Resources are supplied for a given demand in order of absolute advantage.
Proof. We show that if a resource is supplied for a given demand then a lower net cost resource
will not subsequently be supplied for that demand. Thus by definition, resources supplied for a
given demand must be in order of absolute advantage. Let qaj(t1) > 0 and waj > wbj for resources
a,b Є R and demand j Є U at time t1 Є (t0,∞). From (6),
waj + λ a ( t1 ) = p j ( t1 ) ≤ wbj + λb ( t1 )
(10)
15
which since waj > wbj implies that λa < λb. From (5), λ&a < λ&b and so the left hand side of (10)
increases at a slower rate than the right hand side. Thus waj + λ a ( t ) < wbj + λb ( t ) for all t Є
(t1,∞).Then from (6), pj(t) ≤ λa(t) + waj < λb(t) + wbj for all t Є (t1,∞) and so qbj(t)=0 for all t Є
(t1,∞). ■
Proposition 7 does not say that all resources will be supplied for each demand but that of those
resources that are supplied, their use will be in strict order of absolute advantage. In this sense,
the Herfindahl Principle of “least cost first” is preserved within each demand. In the special case
of a single demand, Proposition 7 reduces to the Herfindahl principle.
Proposition 8. Resources of the same resource class are extracted in order of decreasing grade.
Proof. We show that if one resource is being extracted, then a higher grade of the same resource
will not subsequently be extracted. Thus resources within the same class are extracted in order of
decreasing grade. Let Q& a ( t1 ) > 0 and Q& b ( t1 ) = 0 for resources a,b Є R at time t1 Є (t0,∞) where
resource b is a higher grade of the same resource class as resource a. By the last inequality, from
(4) there exists c Є U such that qac(t1) > 0. From (6),
wac + λ a ( t1 ) = p c ( t1 ) ≤ wbc + λb ( t1 )
(11)
which since wbc < wac from the definition of higher grade, implies that λa(t1) < λb(t1). From (5),
λ&a < λ&b , the left hand side of (11) increases at a slower rate than the right hand side, and so wac
+ λa(t) < wbc + λb(t) for all t Є (t1,∞). Since wbj – waj is constant for all j Є U (from the
definition of resource class), this implies that waj + λa(t) < wbj + λb(t) for all t Є (t1,∞) and j Є
16
U. Combining with (6) yields pj(t) ≤ waj + λa(t) < wbj + λb(t) which implies that qbj(t)=0 for all t
Є (t1, ,∞) and j Є U. From (4), Q& b ( t ) = −
∑q
j∈U
bj
( t ) = 0 for all t Є (t1, ,∞). ■
If there is a single resource class, all demands can be aggregated into one composite demand and
Proposition 8 reduces to the Herfindahl Principle. Since the proposition demonstrates that
deposits within a resource class will be extracted in strict order of grade, we can aggregate
resource grades and consider the resulting composite resource as having an extraction cost
function that increases with cumulative extraction. This provides a microeconomic foundation for
resources with rising, cumulative extraction cost functions, used frequently in the literature (e.g,
Heal, 1976).
Since demand is positive at all prices, there will always be a resource available for each demand.
The following provides a condition under which all resources except one will be exhausted.
Proposition 9. A resource that is strictly inferior, i.e. dominated by all other resources, will
eventually be used exclusively for all demands.
Proof. Let resource a Є R be strictly inferior. From Proposition 5, λa < λi for all i Є R – {a}. Since
scarcity rents rise exponentially, there exists a time ta Є (t0,∞) such that waj + λa(t) < wij + λi(t)
for all i Є R – {a}, j Є U, t Є (ta,∞). From (6), qij(t)=0 for all i Є R – {a}, t Є (ta,∞) and so qaj(t) >
0 for j Є U, t Є (ta,∞). ■
Given that scarcity rents are ordered, a strictly inferior resource will have the lowest scarcity rent
in all stages. A strictly inferior resource that is unlimited in quantity thus corresponds to the
notion of “backstop resource” in the resource economics literature. At the other end of the
spectrum, it is natural to ask if a single resource could be used exclusively for all demands at time
17
t0. The next proposition demonstrates that if there is a resource that is relatively cheap and
plentiful, it will be used exclusively, i.e., for all demands at the beginning of the extraction
program.8
Proposition 10. A resource a Є R that is universally dominant will be used exclusively at time t0 if
s j + waj
∑ ∫ D ( p ) /( p − w
j∈U s0 + waj
j
aj
)dp < rQa ( t 0 ) where
s0 = min {wij - waj | i Є R - {a}, j Є U}, and
sj = max {wij - waj | i Є R, j Є U.
Proof. See Appendix 1.
On the other hand, if each resource has absolute advantage in one demand, then no resource can
supply all demands in any stage. This is shown below for the case of an equal number of
resources and demands (i.e., m=n), sometimes referred to as “even” models.9
Proposition 11. When each resource has absolute advantage in an “even” model, no resource
can supply all demands.
Proof: Without loss of generality, re-label all demands from 1,..,m so that resource i has absolute
advantage in demand i, i=1,..,m, that is, wii < w ji ,∀j ≠ i . Let resource 1 be used for all demands,
for some t Є I where I ⊂ ( t 0 , ∞ ) . Then define φ j1 j ( t ) = p1 j ( t ) − p jj ( t ), j ∈ {U \ 1}. That is,
8
An intuitive explanation for this result is provided for the 2 x 2 case in section 3.
9
The even-odd terminology follows extensions of Heckscher-Ohlin trade theory to higher dimensions. For
an overview, see Bhagwati, et al., (1998).
18
φ j1 j is the price differential between resource 1 and resource j in demand j. Since resource 1 is
used for all demands in this interval, φ j1 j ( t ) = ( λ1( t ) − λ j ( t )) + ( w1 j − w jj ) < 0 . This implies
that λ1( t ) < λ j ( t )∀j ≠ 1 since w1 j > w jj . By (5), this inequality holds over the entire time path.
Also, resource 1 is used for all demands, so it is cheaper relative to any other resource k in
demand j, i.e., φ j1k ( t ) < 0∀t ∈ I . Since resource 1 has the lowest shadow price,
φ& j1k ( t ) = λ&1( t ) − λ&k ( t ) = r( λ1( t ) − λk ( t )) < 0∀t ∈ [ t0 ,∞ ),k ∈ R , j ∈ {U \ 1}. That is, resource
1 will always be cheaper than other resources for every demand. No other resource will ever be
used subsequently, contradicting Lemma 1. ■
The above proposition is independent of the stock sizes of the resources. That is, however
abundant a resource may be, it will never be the exclusive supplier for all demands. As resources
get exhausted, one resource may become universally dominant and thus become the exclusive
supplier.
Ricardian Comparative Advantage
In the following, we develop notions of comparative advantage and show that this taxonomy
plays an important part in characterizing the sequence of resource extraction. In particular, a
resource with universal dominance and universal comparative advantage will always be used
first. We thus generalize the Herfindahl notion to the case with multiple demands.
19
Definition. Resource a Є R has a pairwise comparative advantage in use k over resource b Є R if
wbk − wak > wbj − waj , some j Є {U \ k}.10
Definition. Resource a Є R has comparative advantage in use k over resource b Є R if
wbk − wak > wbj − waj , all j Є {U \ k}.11
Definition. Resource a Є R has universal comparative advantage in use k if it has a comparative
advantage over all other resources, i.e. if wik − wak > wij − waj , all j Є {U \ k}, i Є {R \ a}.
These definitions parallel the notions of absolute advantage presented earlier. If oil has pairwise
comparative advantage relative to coal in transportation, then the net cost differential between oil
and coal in transportation exceeds the net cost differential in some other demand.12 Comparative
10
The last inequality implies that
waj − wbj < wak − wbk . That is, the definition of pairwise comparative
advantage is symmetric, i.e, resource b has a pairwise comparative advantage in use j over resource a.
However it is easy to check that it is also transitive, i.e., if resource a has pairwise (k and j) comparative
advantage in use k relative to resource b, and b has pairwise (k and j) comparative advantage in use k
relative to resource c, then resource a has pairwise (k and j) comparative advantage in use k relative to
resource a.
11
Similar distinctions between the three types of comparative advantage may be useful in international
trade theory as well.
12
If distinct resources have absolute advantage in specific demands, it leads to pairwise comparative
advantage. Suppose resource a (b) has absolute advantage over resource b (a) in demand j (k). Then waj <
wbj and wbk < wak which upon subtracting inequalities yields wbk - wak < wbj - waj, implying that resource a
(b) has pairwise comparative advantage relative to resource b (a) in use j (k). From Proposition 12 below,
no resource can be an exclusive supplier under pairwise comparative advantage.
20
advantage of oil in transportation, say relative to coal implies that the net cost differential
between oil and coal is higher in transportation than in every other demand. Comparative
advantage over another resource implies pairwise comparative advantage over all demands.
Universal comparative advantage of oil in transportation implies that the net cost differential
between oil and all other resources is higher in transportation than in every other demand.
Universal comparative advantage in a given use implies comparative advantage relative to all
other resources. This taxonomy allows us to develop criteria for ranking resources by their
comparative advantage.
The following result specifies that if two resources have pairwise comparative advantage, they
cannot simultaneously be extracted for the uses wherein the other resource has the advantage.
This result limits the set of solutions, as we will see in the 2 x 2 case.
Proposition 12. If resources a and b have pairwise comparative advantage relative to one another
in j and k respectively, then a cannot be used in k while b is used in j; a, b Є R; j, k Є U.
Proof. Let qak(t) > 0, qbj(t) > 0 for t Є I where I ⊂ ( t 0 , ∞ ) . By the definition of pairwise
comparative advantage, wbk - wbj < wak - waj. Then from Proposition 2,
p k ( t ) = wak + λ a ( t ) ≤ wbk + λb and p j ( t ) = wbj + λb ( t ) ≤ waj + λ a ( t ) . Subtracting the
inequalities yields λb + wbk − ( λb + wbj ) ≥ λ a + wak − ( λ a + waj ) so that
wbk − wbj ≥ wak − waj which is a contradiction. ■
Next we show that both universal dominance and universal comparative advantage are sufficient
to ensure that a resource is used before any other resource:
21
Proposition 13. A resource with universal dominance and universal comparative advantage in
demand k must be used exclusively in that demand.13
Proof: Let resource a have universal dominance and universal comparative advantage in use k.
Let another resource b ∈ R supply demand k over an interval I 1 = [ t1 ,t 2 ) ⊂ [ t 0 , ∞ ) . Then wak+
λa(t) = pk(t) ≥ wbk+ λb(t) ∀t ∈ I 1 so that 0 < wbk - wak ≤ λa(t) – λb(t), where the first inequality is
from universal dominance of a in k. However, over another interval I 2 = [ t3 ,t4 ) ⊂ ( t2 ,∞ ) ,
resource a must be used for some demand, say j. Then pj(t)=waj+ λa(t) ≤ wbj+ λb(t) ∀t ∈ I 2 which
implies that λa(t) - λb(t) ≤ wbj - waj ∀t ∈ I 2 . Since λa(t)≥ λb(t) from above, the shadow prices
must diverge by (5), hence λ&a ( t ) − λ&b ( t ) ≥ 0 and by the definition of the intervals I1 and I2,
{ λ a ( t ) − λb ( t ) t ∈ I 1 } ≤ { λ a ( t ) − λb ( t ) t ∈ I 2 } . Consolidating, we get
wbk − wak ≤ { λ a ( t ) − λb ( t ) t ∈ I 1 } ≤ { λ a ( t ) − λb ( t ) t ∈ I 2 } ≤ wbj − waj . This inequality
must hold for any resource b and demand j, i.e., wik − wak ≤ wij − waj , which contradicts the
definition of universal comparative advantage of resource a in use k. ■
13
Our notion of pair-wise comparative advantage is equivalent to the definition of comparative advantage
put forward by Gaudet, Moreaux and Salant (2001). Comparative advantage, in their model determines,
given an arbitrary use profile, which city will switch first to a higher cost landfill site. A city (in our case,
demand) may switch to a more costly site (in our analogy, use a more costly resource) if it has comparative
advantage in that demand. However, their definition of comparative advantage focuses on resource
switching and does not predict which resource will be used first. Our (stronger) definition of comparative
advantage suggests that if a resource with universal dominance and universal comparative advantage exists,
it is automatically picked as the exclusive supplier at the beginning of the planning horizon.
22
A resource with universal dominance and universal comparative advantage in demand k must be
used exclusively in that demand at the beginning of the planning horizon. Universal dominance
and universal comparative advantage are sufficient for the above result to hold. They establish a
clear ordering of resource extraction across resource classes.
The two definitions of absolute advantage – dominance and universal dominance – are equivalent
when there are only two resources. In this case, all three definitions of comparative advantage are
also equivalent. This is because, with two demands, if resource a has dominant comparative
advantage in use k relative to resource b, the cost differential between resource a and b in demand
k is higher than in all other demands. If the number of demands is only two, there is only one
“other” demand. Thus a has universal comparative advantage over b in use k. The argument for
the other equivalencies is similar.14 In the Herfindahl case when the number of demands is unity,
the definitions of comparative advantage reduce to the following:
Definition. Under one demand, resource a has universal comparative advantage if
wik − wak > 0,i ∈ { R \ a }.
That is, a resource has comparative advantage when its net cost is lower than that of all other
resources. Then Proposition 13 reduces to the Herfindahl Principle, i.e., the resource with the
lowest net cost must be used exclusively. Furthermore, the sequence of extraction must be
according to the “least cost first” principle. Unlike in the one demand case (except when
resources have equal net costs), a resource with universal comparative advantage may not exist
under multiple demands.
14
In any two-resource model, comparative advantage and universal comparative advantage are equivalent.
All three definitions are distinct in models with dimensionality m > 2, n >2. Precise results on their
relationships in different dimensions can be explored in future work.
23
Absolute and Comparative Advantage: Polar Cases
The role of absolute and comparative advantage in the general case may be illuminated by two
polar extremes: (i) for each resource, conversion costs are the same across end uses (ii) each
resource enjoys a symmetrical comparative advantage in each end use.
For (i), with equal conversion costs across demands for each resource, we have
vij = vik ∀j, k ∈ R, i ∈ U . Then wij = ci + vij = ci + vik = wik , so that for any two resources a and b,
wbj − waj = cb + vbj − ( ca + v aj ) = cb + v bk − ( ca + v ak ) = wbk − wak ∀j, k ∈ U , since vij = vik . Hence
by definition, all resources are of the same resource class. Let us re-label the resources in order of
increasing net cost, i.e., w1 j < w2 j < .. < wmj . Then by Proposition 6, λ1 > λ2 > .. > λm . By
Proposition 8, resources will be extracted in order of decreasing “grade”, with resource 1 in the
initial stage and m at the end. The sequence of extraction will follow the Herfindahl Principle,
i.e., each resource will supply all demands exclusively, until it is exhausted and the next higher
net cost resource is employed.
For (ii), consider the m x m model, in which each resource has universal comparative advantage
in a given demand. Suppose demands are identical and stocks of each resource are equal. Without
loss of generality, the demands are re-labeled such that resource i has universal comparative
advantage in demand i, i ∈ I = {1,.., m}. Suppose w11 = w22 = .. = wmm < w1i1 = w2i2 = .. = wmim ,
where i1 ∈ {I \ 1},i2 ∈ {I \ 2},..,im ∈ {I \ m}. That is, the net costs of all resources are equal in their
respective demands and higher (and equal) for all other demands. In this perfectly symmetrical
world, since there is no distinction between resources except by comparative advantage, and
demands are identical, λ1 = λ2 = .. = λm . By Proposition 2, each resource will be used exclusively
24
in the demand in which it has comparative advantage. Because of perfect symmetry, each
resource will be exhausted at infinity. To summarize, in the first polar extreme, resource use is
determined entirely by absolute advantage; in the second, entirely by comparative advantage.
More generally, resource use will reflect a trade-off between both forces.
Comparative Dynamics
In order to investigate the comparative dynamics properties of the m x n model, we can define the
value function in (1)-(3) as
∞
V ( Qi ( t 0 )) = ∫ B( Qi ( t 0 ),t )dt , where
(12)
t0
B( Qi ( t 0 ),t ) = e
− rt
Z
[ ∑ ∫ D −j 1 ( x )dx − ∑ ∑ wij qij ( t )]
j∈U 0
i∈R j∈U
represents the discounted benefits from extraction at any given instant of time t. To relate the
change in resource scarcity rents to the change in the aggregate resource stocks, we need to
establish the differentiability of the value function V ( Qi ( t 0 ),t ) . Benveniste and Scheinkman
(1979) have shown that under fairly general conditions as in our case, the value function V(·) is
once differentiable. In particular, we invoke their Corollary 1, where the optimal control is a
piecewise continuous function of time. In our case, the control functions are discontinuous since
there may exist intervals I such that qij(t)=0, t Є I and I ⊂ ( t 0 , ∞ ) . By Proposition 7, and because
we have only a finite number of demands and resources, there can only be a finite number of such
intervals. At the switch points between these intervals, the state variables may not be
differentiable and the control functions may be discontinuous. In Appendix 2, we check that (12)
satisfies the assumptions of their Corollary 1. This gives the following result:
25
∂V ( Qi ( t 0 ),t )) / ∂Qi ( t 0 ) = λi ( t 0 ) ,
which implies that the initial scarcity rent is the derivative of the optimal value function with
respect to the initial stock of the resource. In another paper, under more general conditions,
Benveniste and Scheinkman (1982) have shown that the value function is concave. That is, if
Qi ( t 0 ),Q̂i ( t 0 ) are two different initial stocks of any resource i, then
( Q( t 0 ) − Q̂( t 0 )).(( λ ( t 0 ) − λˆ ( t 0 )) ≤ 0 , where λ ( t 0 ),λˆ ( t 0 ) represent the corresponding
vector of scarcity rents at time t0. Thus if Q̂i ( t 0 ) > Qi ( t 0 ) , e.g., representing an exogenous
increase in the initial stock of resource i, then λˆ i ( t 0 ) ≥ λi ( t 0 ) , provided that all other stocks
remain unchanged. We can then state the following result:
Proposition 14. An exogenous increase in the initial stock of a resource causes a reduction in its
scarcity rent.
This implies that in the limit, if a given resource i is inexhaustible,
yields
lim
Qi ( t 0 )→∞
lim λi ( t ) = 0 which
Qi ( t 0 )→∞
pij ( t ) = wij + lim λi = wij . If all other resources are limited in quantity, the ith
Qi ( t 0 )→∞
resource is a backstop resource. In section 3, we examine the role of resource abundance for the 2
x 2 case.
3. Characterization of the 2 x 2 Case
We now consider the simplest possible setting with multiple demands, that of two demands and
two resources. In this case, dominance implies universal dominance and pairwise comparative
advantage is equivalent to comparative and universal comparative advantage. Without loss of
26
generality, we assume that the demands are denoted by electricity and transportation, and the
resources are oil and coal. There are only two possible cases to consider: (i) one resource is
dominant, i.e., has absolute advantage in both demands and (ii) when each resource has absolute
advantage, i.e, has absolute advantage in some demand. Again without loss of generality, we
consider the following two cases: (i) oil is a dominant resource, and (ii) oil has absolute
advantage in transportation and coal in electricity.15
Oil is dominant
We can simplify notation by letting wOE and wOT (wCE and wCT) denote net costs of oil (coal) to
electricity and transportation, respectively. Note that dominance for oil implies that
wOE < wCE and wOT < wCT . Define k = ( wCT − wOT ) + ( wOE − wCE ) . Then oil has
comparative advantage in transportation (electricity) if k > (<) 0. The following propositions
provide sufficient conditions for oil to be used at the beginning for both uses, when it is
dominant:
Proposition 15. If oil is dominant and has comparative advantage in transportation, and the
wCT
condition
∫
wCT − k
DT ( p )
dp < rQO ( t o ) holds, then oil must be used for both uses at the
p( t ) − wOT )
beginning. In the second stage, oil is used for transportation and coal for electricity. In the third
stage, coal is used for both uses.
Proof. See Appendix 1.
15
A similar example is used in Chakravorty and Krulce (1994), albeit under the stricter assumption that oil
is the lower cost resource for both demands and has a comparative advantage in transportation.
27
Proposition 15 suggests that only oil will be used at the beginning if it is abundant, the discount
rate is high, the net cost of coal in transportation is high (since demand is downward sloping), the
net cost of oil in transportation is low, the demand for transportation is low, and k is low. Recall
that the magnitude of k denotes the degree of comparative advantage of oil in transportation over
coal in electricity. The higher the value of k, the less likely it is that oil will be used for electricity
in the beginning stage. That is, comparative advantage of oil in transportation makes it less likely
that oil will supply both demands in the initial stage. Note that in this stage, oil is used for
electricity even though coal has comparative advantage in that demand. The effect of dominance
is stronger than that of comparative advantage. The solution is shown in Fig.1. Only oil is used
until time t1, followed by coal in electricity and oil in transportation until time t2, when oil is
exhausted. In the third stage, the strictly inferior resource, coal supplies both demands.
However, even if oil is dominant, i.e., has absolute advantage over coal in both demands, it may
have comparative advantage in electricity, not in transportation. The following result shows that
stage 2 will then be different. It is the mirror image of Proposition 15, so the proof is not given
separately.
Proposition 16. If oil is dominant and has comparative advantage in electricity, and the condition
wCE
∫
wCE − k
DE ( p )
dp < rQO ( t o ) holds, then oil must be used for both uses at the beginning. In
p( t ) − wOE )
the second stage, oil is used for electricity and coal for transportation. In the third stage, coal is
used for both uses.
In this case k = ( wCT − wOT ) + ( wOE − wCE ) < 0. The solution is exactly as in Fig.1 except that
in stage 2, it is oil that is now used for electricity and coal in transportation. Both solutions are
28
summarized in Table 2(a,b). A reversal is obtained in stage 2, and the sequence of extraction in
this stage is strictly according to comparative advantage, as implied by Proposition 12. Since the
solutions in Proposition 15 and 16 are unique, we get the following corollary:
Corollary. When both resources are extracted simultaneously in a 2 x 2 model, resources must be
extracted for the demand in which they have comparative advantage.
Since the stages of resource use are determined according to the principle of least-shadow-pricefirst, Fig.2 shows the solution described in Proposition 15 in terms of shadow price differences.
The functions ΦE(t) and ΦT(t) denote the price of oil net of coal in electricity and transportation,
respectively.16 When ΦE(t) < 0, the price of oil is lower than that of coal; hence oil is used for
electricity. When ΦE(t) > 0, coal becomes cheaper and substitutes for oil. As shown in the proof
of Proposition 15, φ E ( t ) > φ T ( t ). When k=0, φ E ( t ) ≡ φ T ( t ) and the middle stage is
eliminated. Oil is used exclusively at the beginning followed by coal at the end. The transition
from oil to coal happens in both sectors simultaneously. Thus, the Herfindahl result of “least cost
first” is obtained as a special case when oil has absolute advantage in transportation and coal in
electricity, but no resource has comparative advantage. The cost advantage of oil and coal in their
respective demands cancel each other.17 The following proposition relates the length of the
second stage to the magnitude of comparative advantage, denoted by k:
φ E ( t ) = pOE ( t ) − pCE ( t )
φ T ( t ) = pOT ( t ) − pCT ( t ) .
16
That is,
17
The model of Chakravorty and Krulce (1994) is obtained by substituting
and
wOE = wOT = cO , wCE = cC ,
wCT = cC + z , so that k = wCT − wCE = z > 0. Their Proposition 1 is a special case of our
Proposition 15.
29
Proposition 17. If oil is dominant and has comparative advantage in transportation, and the
inequality stated in Proposition 15 holds, the length of the second stage increases with k and

decreases with ( wCE − wOE ) and r. It is given by t 2 − t1 = ln( 1 +

wCE

k
) / r , where t1(t2)
− wOE 
is the switch point from stage 1(2) to stage 2(3).18
Proof: Since ϕ E ( t ) = p OE ( t ) − pCE ( t ) = ( λO ( t 0 ) − λC ( t 0 ))e rt + ( wOE − wCE ),

wCE − wOE
 λ o ( t 0 ) − λC ( t 0
ϕ E ( t1 ) = ( λO ( t O ) − λC ( t O ))e rt + ( wOE − wCE ) = 0, so that t1 = ln
1

 / r.
)
wCT − wOT 
 / r . Subtracting t1 from t2 and some
 λo ( t 0 ) − λC ( t 0 ) 

Similarly, ϕ T ( t 2 ) = 0 implies t 2 = ln
algebraic manipulation yields the result. ■
The above proposition suggests that the higher the magnitude of comparative advantage the
longer the stage where both resources are extracted simultaneously. The higher the absolute
advantage of oil in electricity, the smaller the length of the stage in which there is joint extraction
of the two resources. This is intuitive because if oil has a high degree of absolute advantage in
electricity, it is likely to be used in stage 1 and not in stage 2. This will decrease the use of oil for
transportation in stage 2. Thus, dominance of oil leads it to be used in stage 1, while comparative
advantage forces specialization of the two resources in stage 2. A higher discount rate decreases
the length of the second stage. Since oil has absolute advantage in both resources, a higher
discount rate implies that profits from the use of oil in transportation in stage 2 are discounted
more heavily, reducing its duration. Conversely, as we saw in Proposition 15, a higher discount
rate suggests that oil is better used in stage 1 for generating profits earlier in the time horizon.
18
An analogous result could be obtained for the case of Proposition 16.
30
Oil has Absolute Advantage in Transportation and Coal in Electricity
We now consider the case wherein both resources have one absolute advantage each. We can then
state the following:
Proposition 18. When oil has absolute advantage in transportation and coal in electricity, there
are only two possible solutions, each with two stages. The first stage is the same in both solutions
-- oil is used for transportation and coal for electricity. In the second stage, either oil or coal is
used exclusively for both uses.
Proof: Let λO ( t 0 ) > λC ( t 0 ) . Then from the proof of Proposition 9, coal will be used for both
uses in the terminal stage. But φ E ( t ) = ( λO ( t ) − λC ( t )) + ( wOE − wCE ) > 0 and is increasing
in t. If φ T ( t 0 ) = ( λO ( t 0 ) − λC ( t 0 )) + ( wOT − wCT ) > 0 then since φ&T ( t ) > 0 , oil will never
be used, violating Lemma 1. Thus φ T ( t 0 ) < 0 . In the first stage, oil is used for transportation and
coal for electricity, followed by coal in both uses. The proof for the case when λO ( t 0 ) < λC ( t 0 )
is similar and not repeated.19 ■
By Proposition 14, if coal is sufficiently “abundant” relative to oil, its scarcity rent will fall. In
that case, λC ( t 0 ) < λO ( t 0 ) , so that the terminal stage uses coal for both demands. This can be
seen graphically in Fig.2 with one modification: ΦE(t) is now everywhere positive so that there is
only one switch point from oil to coal in transportation. Similarly, abundance of oil will imply
that oil will be the terminal resource. Both solutions are summarized in Table 2(c,d). The polar
19
The measure zero event in which
λO ( t 0 ) = λC ( t 0 )
yields a knife-edge equilibrium with complete
specialization – oil is used for transportation and coal for electricity over the entire planning horizon. Both
prices grow at the same rate. The price differentials φ E ( t ),φ T ( t ) are constant functions.
31
case is when the stock of oil is unlimited, and it becomes a (conventional) backstop resource. In
that case, the scarcity rent of oil is zero. The price differentials ΦE(t) and ΦT(t) are decreasing in
time t.
Clean Oil, Dirty Coal: Environmental Taxes in the 2x2 Case
The multiple demand framework allows for the analysis of taxes that are resource or sectorspecific (or both). Again, for simplicity, consider oil to be a “clean’ resource and impose an
exogenous tax on a “dirty” resource (e.g., coal) which is assumed to generate significant negative
externalities. Other cases in which a resource and sector–specific tax (say, on coal in electricity
generation) or only a sector-specific tax (a tax on the transportation sector, for instance, to reduce
local pollution) may be imposed are discussed briefly later in this section. The single demand
framework is inadequate for this purpose because it does not allow fuel substitution across
sectors. We can then state the following result where the term “abundance” implies that the stock
of oil is large enough to satisfy the inequality in Proposition 15:
Proposition 19. If oil is abundant and has absolute advantage in transportation while coal has
absolute advantage in electricity, a tax on coal equal to t C > wOE − wCE will lead to oil being
used exclusively in the beginning.
~ is defined as
Proof: With the above tax, the modified net cost of coal in electricity w
CE
~ = w + t > w + w − w = w . Oil has dominance. By hypothesis,
w
CE
CE
C
CE
OE
CE
OE
k = ( wCT − wOT ) + ( wOE − wCE ) > 0 and is unaffected by the tax. Proposition 15 then implies
that oil will be used exclusively at the beginning. ■
The important difference is that in the one-demand Hotelling model, a tax on the resource reduces
its use. However, when oil has absolute advantage in transportation and coal in electricity, each
32
resource is being extracted in the first stage (Proposition 18). If oil is sufficiently abundant, a
large enough tax on coal will lead to a shutdown in coal extraction, and both demands will be
exclusively supplied by oil for a period after which, both resources will again be extracted jointly.
The tax allows for a postponement of coal consumption until the future. This policy may be
beneficial if, for example, exogenous research and development (e.g., on clean coal technologies)
over time reduces the negative environmental externalities from coal.20
If a sector-specific tax is imposed, e.g., on all fuels in the transportation sector, the outcome may
depend on which of the two conditions (in Proposition 15 and 16) hold. One interesting case is
when oil is a dominant and abundant21 resource, and has comparative advantage in transportation.
Consider stage 2 of Proposition 15, i.e., oil is being used in transportation and coal in electricity.
A sufficiently large tax on the transportation sector will in effect reduce the magnitude of
transportation demand, so that the inequality in Proposition 15 may now be satisfied, leading to a
switch to stage 1, i.e., oil supplying both forms of energy. Coal will no longer be extracted until
stage 2 is reached again in this perturbed model. That is, a tax on the transportation sector may
induce a shift from coal to oil in electricity. After the tax, oil is extracted for both demands, so
that aggregate consumption of oil may increase.
The implications of the tax may be quite different if the tax is resource and sector-specific, e.g., a
tax on coal consumption in electricity.22 In that case, a tax that is larger than k will result in oil
20
The basic model could be extended to include differential environmental costs of fossil fuels. For
instance, carbon emitted per unit of energy delivered from burning coal, oil and natural gas is
approximately in the ratio 5:4:3. For analytical purposes, these costs could be modeled as extraction costs.
The higher environmental damage from coal will be reflected in a higher net cost of coal in all demands.
21
In the sense that the inequality in Proposition 15 holds.
22
The gasoline tax, popular in many countries, is essentially a resource and sector-specific tax.
33
having comparative advantage in electricity, and coal in transportation. Thus, before the tax, oil is
used in transportation and coal in electricity. After the tax, oil is used in electricity and coal in
transportation.23 The tax causes a complete switching of fuels between the two sectors.
Relative to the predictions of the Hotelling model, taxing the resource or taxing the sector (or a
combination) may have very different implications for the extraction profile and for achieving
pollution targets, a potentially significant policy issue that needs to be considered in detail in
future research. Although we do not have endogenous choice of conversion technology that
converts resources into final demands (e.g., automobiles, power generation equipment), there may
be differential impacts depending on the type of tax imposed. Without a formal model, however,
it is not clear ex-ante how, for example, the adoption of fuel-efficient automobiles will be
affected by these alternative tax mechanisms.
Sectoral Energy Prices
The optimal resource use profile can be used to characterize sector-specific efficiency prices of
energy. 24 Suppose again, that coal is “dirty” and oil is “clean,” and environmental costs of the
resources are incorporated into the net cost so that wCT > wCE > wOE > wOT . This implies that
23
With this tax on coal in electricity, say tCE, k < tCE implies
( wCT − wOT ) − ( wCE + tCE − wOE ) = ( wCT − wOT ) − ( wˆ CE − wOE ) < 0 , where ŵCE is the modified net
cost of coal in electricity. Now oil has comparative advantage in electricity and coal in transportation.
24
More generally, the price of a resource that is efficiently employed in a sector plus its conversion cost in
that sector can be thought of as the shadow price of an intermediate good required for production of the
final good. If no intermediate good (such as energy) can be identified, the conversion cost can be taken as
the non-resource production costs for the final good.
34
oil has a comparative advantage in transportation, i.e., k = ( wCT − wOT ) + ( wOE − wCE ) > 0. 25
Then we can state the following:
Proposition 20. If oil is abundant, then the price of electricity is higher than that of transportation
energy in the beginning and lower in the end. Within each stage, the sectoral energy prices grow
at the same rate and the price differential is monotone decreasing over time.
Proof: By Proposition 15, the abundance of oil implies it is used exclusively at the beginning, and
coal at the end. In the beginning stage,
p E − pT = ( λO + wOE ) − ( λO + wOT ) = wOE − wOT > 0 . In the terminal stage,
p E − pT = ( λC + wCE ) − ( λC + wCT ) = wCE − wCT < 0 . This proves the first part of the
proposition. For the second part, note that pE and pT grow at the same rate in the first and last
stages. In the middle stage, p& T = λ&O = rλO ( t ) > rλC ( t ) = λ&C = p& E . Thus pE - pT declines. ■
The two price paths are shown in Fig. 3. In the first stage, pE is higher than pT and both grow at
the same rate. In the second stage, the price of transportation energy rises at a faster rate than
electricity, and cuts the latter from below. In the terminal stage, both prices again grow at an
equal rate.26 Since pE - pT declines over time, it follows that if the price of electricity is lower in
25
The assumption of clean oil, dirty coal used here is stronger than that of oil with dominance and
comparative advantage in transportation. The former provides an empirical justification for the latter more
general set of assumptions.
26
These sectoral price paths are purely a result of the relative ordering of net costs. The relative magnitude
of demands in the two sectors plays a role only to the extent that the condition for Proposition 15 is
satisfied. If the relative ordering of
wOE , wOT ( wCE , wCT ) were indeterminate, then the relative ordering
of p E , pT would also be indeterminate. However, the price differential is driven by comparative advantage
35
the beginning then it must be lower for all subsequent periods. Notice that when oil has
dominance and is used exclusively in the beginning, the inequality
wOE − wOT > wCE − wCT holds, hence p E − pT is always lower in the terminal stage relative to
the initial stage. Thus a general consequence of comparative advantage in the 2 x 2 model is:
Corollary: If a dominant resource is used at the beginning, the sectoral price differential must
always decrease over time.
4. Concluding Remarks
In a model of multiple resources that satisfy multiple demands, optimal resource use is
determined by two forces. Specialization of resources according to demand is driven by Ricardian
comparative advantage. The order of resource use over time is determined by Ricardian absolute
advantage, whereby resources with the lowest extraction and conversion costs tend to be used
first. Each principle partially masks the other, and only in polar cases do the pure forms of each
principle emerge. In one such polar case, wherein conversion costs are independent of demand,
resources will be used in order of absolute advantage, i.e. least cost first. At the other extreme, in
a m x m model, if each resource enjoys an exactly symmetrical universal comparative advantage
in one demand and all demands and resource supplies are similarly symmetrical, then each
resource will specialize in one and only one demand until simultaneous exhaustion.
In the general case, results are more limited, but the same tendencies prevail. Within demands,
resources are used in strict order of absolute advantage. Universally dominant resources are
initially employed in all demands, provided that they are sufficiently abundant. Strictly inferior
and still declines monotonically over time. It is easy to check that it holds even if there was no initial stage
with exclusive use of oil.
36
resources are exclusively used in the last stage of resource use. In the polar case, an unlimited
quantity of a strictly inferior resource is a backstop resource. With an equal number of resources
and demands, if each resource has absolute advantage in a given demand, no resource could
exclusively supply all demands. As this case converges towards perfect symmetry, the use profile
converges to a single stage solution, wherein each resource is used only according to comparative
advantage. On the other hand, where resources can be ranked according to dominance (one
resource with universal dominance, the next with universal dominance over the rest), then
resource use converges to least-cost-first as conversion costs across demands become
decreasingly disparate.
With only two resources and two demands, either each resource has an absolute advantage in one
end use or one resource is dominant. The possible sequences of resource use are characterized
according to resource scarcity and the costs of extraction and conversion, thus highlighting the
differentiated roles of absolute and comparative advantage as well as resource abundance. When
a resource has a dominant absolute advantage and is sufficiently abundant, it is used exclusively
at the beginning, contrary to the principle of comparative advantage. For example, oil may be
used for electricity initially even though coal has comparative advantage in that demand.
However, if each resource has absolute advantage in a given use, then resources are always used
according to comparative advantage, and no resource may be an exclusive supplier, except in the
terminal stage.
Taxes on a dirty resource or on a given sector may induce significant substitution effects,
including substitution towards sectors not being taxed, or a complete switch in resource use
between sectors. For example, a resource and sector-specific tax (e.g., a gasoline tax) may shift
comparative advantage such that oil is used in electricity and coal in transportation. The
implication is that reduction of oil consumption (e.g., to reduce dependence on imports) or of coal
37
consumption (e.g., to reduce carbon emissions) may entail taxing those resources in all uses.
More generally, evaluating the long-run consequences of alternative energy tax proposals requires
calculation of the full energy use trajectory for each of the alternatives.
The explanatory power of the neo-Ricardian theory described above can be enhanced by
extending the model to include demand shifts, technological change, policy distortions, and
transportation costs. For example the transition from whale oil to petroleum in the late nineteenth
century has been attributed to the invention of the kerosene lantern. The transition from coal to oil
in the late 19th and early 20th centuries is said to be due to advantages in transportation, the advent
of the automobile, and technological improvements that lowered extraction and conversion
costs.27 Further research is also needed to characterize higher dimensional models more fully.
One can envision a similar theory being developed for explaining patterns of international trade.
Dynamic comparative advantage has long been linked to changing endowments of factors.
Inasmuch as resource depletion and capital accumulation are two sides of the same capitaltheoretic coin, one expects the trade-off between dynamic and intersectoral specialization to carry
over to neoclassical trade theory.
27
See e.g., Rhodes (2002). The extent to which these changes were induced may also be the subject of
further investigation.
38
Appendix 1
PROOF OF PROPOSITION 1. We adopt Farzin’s (1982) proof of Theorem 15 in Seierstad and
Sydsaeter (1987, Chapter 3) to show that a unique optimal solution exists to (1)-(3) and Theorem
13 to show that the necessary conditions are also sufficient. Define f 0 ( d ij ( t ),t i ∈ R , j ∈ U ) as
the integrand of (1) and f i ( d ij (t ) j ∈ U ) = −
∑d
ij ( t ), i ∈ R .
Then we need to prove the following
j∈U
statements:
1. The functions f 0 ( d ij ( t ),t i ∈ R , j ∈ U ) and f i ( d ij ( t ) j ∈ U ),i ∈ R are continuous.
Proof: By inspection.
2. The functions d ij ( t ),i ∈ R , j ∈ U are bounded.
Proof: Since D j ( p ) is bounded by assumption for any given j, d ij ( t ) is bounded, each i ∈ R .
This is true for all j ∈ U . ■
{
}
3. For any admissible path d ij ( t ) i ∈ R , j ∈ U , there exists a piecewise continuous function
∞
φ 0 ( t ) with ∫ φ 0 ( t )dt < ∞ such that f 0 ( d ij ( t ),t i ∈ R , j ∈ U ) ≤ φ 0 ( t ) .
t0
∑ dij ( t )
∞
Proof: Since
∫ D j ( p )dp < ∞ , j ∈ U ,
i∈R
∫ D ( p )dp is bounded for all
j
t0
j ∈ U . Since all other
t0
terms in the integrand of (1) are bounded, e rt f 0 ( d ij ( t ),t i ∈ R , j ∈ U ) < K where K is some
upper bound. Let φ 0 = Ke
− rt
∞
so that ∫ φ 0 ( t )dt = K / r < ∞ . Then
t0
f 0 ( d ij ( t ),t i ∈ R , j ∈ U ) ≤ Ke − rt = φ 0 ( t ). ■
39
{
}
4. For any admissible d ij ( t ) i ∈ R , j ∈ U , there exists piecewise continuous functions
∞
φ i ( t ),i ∈ R with ∫ φ i ( t )dt < ∞ such that f i ( d ij ( t ) j ∈ U ) ≤φ i ( t ),∀i ∈ R.
t0
Proof. Let φ i ( t ) = −Q& i ( t ) . Then
∞
∞
t0
t0
∫ φ i ( t )dt = − ∫ Q& i ( t )dt = − lim Qi ( t ) + Qi ( t 0 ) ≤ Qi ( t 0 ) since Qi ( t ) ≥ 0 from (2).
t →∞
Now d ij ( t ) ≥ 0 implies from (3) that Q& i ( t ) ≤ 0 hence
f i ( d ij ( t ) j ∈ U ) = − ∑ d ij ( t ) = Q& i ( t ) = −Q& i ( t ) = φ i ( t ) . This holds for all i ∈ R. ■
j∈U
{
}
5. For any admissible d ij ( t ) i ∈ R , j ∈ U , there exist non-negative functions a(t) and b(t) such
that
f1, ( d ij ( t )), f 2 ( d ij ( t ),... ≤ a( t ) Qi ( t ),Q2 ( t ),... + b( t ).
Proof. Since f i ( d ij ( t )) = −
2
∑d
j∈U
ij
( t ) by definition, it is bounded by statement 2, ∀i ∈ R. Thus
2
the norm f1 , f 2 ,... = ( f 1 + f 2 + ...)1 / 2 is bounded. Let b(t) be this bound and put a(t)≡0. ■
6. The function f 0 ( d ij ( t ),t i ∈ R , j ∈ U ) is concave for all t.
D ′j ( p ) < 0∀j ∈ U and linearity of the rest of the terms in (1) imply concavity of f 0 ∀t . ■
PROOF OF PROPOSITION 10. Suppose that qac(t0)=0 for some c Є U. Then since demand is
positive, there exists b Є R such that qbc(t0) > 0. Then from (6),
wbc + λb ( t 0 ) = pc ≤ wac + λ a ( t 0 ) which since wbc – wac ≥ s0 implies that
λ a ( t 0 ) ≥ s 0 + λb ( t 0 ) which from (8) yields
λ a ( t ) = λ a ( t 0 )e rt ≥ ( s 0 + λb ( t 0 ))e rt = s 0 e rt + λb ( t ) .
40
(A1)
Since resource a is universally dominant, sj>0 for j Є U. Let t̂ j = log( s j / s 0 ) / r for j Є U so
that
s0ert > sj for t > t̂ j and j Є U.
(A2)
Then from (A1), (A2) and the definition of sj,
waj + λ a ( t ) ≥ waj + s o e rt + λb ( t ) > waj + s j + λb ( t ) ≥ wbj + λb ( t ) for t > t̂ j and j Є U. So
from (6),
qaj(t)=0 for t > t̂ j , j ∈ U .
(A3)
Let γj(t) = waj + s0ert for j Є U so that
γ j ( t 0 ) = waj + s 0 ,γ j ( t̂ j ) = waj + s j , and γ j ′ ( t ) = rs 0 e rt = r( γ j ( t ) − waj ) for j Є U. (A4)
From (A1) and since λb(t)≥ 0,
waj + λ a ( t ) ≥ waj + s 0 e rt + λb ( t ) ≥ waj + s 0 e rt = γ j ( t ) for j Є U.
(A5)
From (6),
p j ( t ) = waj + λ a ( t ) ⇒ D j ( waj + λ a ( t )) = D j ( p j ( t )) ≥ q aj ( t ) for j Є U and
p j ( t ) < waj + λ a ( t ) ⇒ q aj ( t ) = 0 ≤ D j ( p j ( t )) for j Є U, and therefore
D j ( waj + λ a ( t )) ≥ q aj ( t ) for j Є U.
(A6)
Thus
s j + waj
∑ ∫
j∈U s0 + waj
Dj( p )
( p − waj )dp
=∑
γ j ( t̂ j )
∫
j∈U γ j ( t 0 )
Dj( p )
( p − waj )dp
t̂ j
∞
j∈U t 0
j∈U t 0
t̂ j
=r ∑ ∫ D j ( γ ( t ))dt
j∈U t 0
≥ ∑ ∫ D j ( waj + λ a ( t ))dt ≥ r ∑ ∫ q aj ( t )dt = rQa ( t 0 )
(A7)
where the change of variables follows from (A4), the first inequality follows from (A5) and that
demand is downward sloping, the second inequality follows from (A3) and (A6), and the last
41
equality follows from (4). Since (A7) contradicts the premise, the supposition is false and so
qac(t0) > 0. Then since c was arbitrary, qaj(t0) > 0 for j Є U. ■
PROOF OF PROPOSITION 15. Define φ E ( t ) = pOE ( t ) − pCE ( t ) and
φ T ( t ) = pOT ( t ) − pCT ( t ) . Then
φ E ( t ) − φ T ( t ) = ( pOE − pOT ) + ( pCT − pCE ) = ( wCT − wOT ) + ( wOE − wCE ) = k . Thus
φ E ( t ) = ( λO + wOE ) − ( λC + wCE ) = ( λO − λC ) + ( wOE − wCE )
= ( λO ( t 0 ) − λC ( t 0 ))e rt + ( wOE − wCE ). Similarly,
φ T ( t ) = ( λO ( t 0 ) − λC ( t 0 ))e rt + ( wOT − wCT ). Suppose λO ( t 0 ) ≤ λC ( t 0 ). Then
φ E ( t ) < 0 ,φ T ( t ) < 0 so that coal is never extracted, contradicting Lemma 1. Thus
λo ( t 0 ) > λC ( t 0 ) and ΦE(t), ΦT(t) are continuous and φ&E ( t ) > 0,φ&T ( t ) > 0. If φ E ( t 0 ) ≥ k ,
then φ T ( t 0 ) = φ E ( t 0 ) − k ≥ 0 so that oil will never be used. Hence φ E ( t 0 ) < k . Let t 0 = t1
where t1 denotes the switch point where electricity supply moves from oil to coal. Then,
φ E ( t 0 ) ≥ 0 . Since ΦE(t) is monotone increasing, φ E ( t ) = pOE ( t ) − pCE ( t ) > 0 for all t ε
(t0,∞) it implies that qOE(t)≡0. Thus λO ( t 0 ) − λC ( t 0 ) ≥ wCE − wOE so that
φ E ( t ) ≥ ( wCE − wOE )( e rt − 1 ). Define t N = (log( 1 +
wCE
k
)) / r . Then for t ≥ tN,
− wOE
φ E ( t ) > ( wCE − wOE )( e rt − 1 ) = k so that φ T ( t ) = φ E − k > 0 , hence qOT(t) ≡ 0 for t ≥ tN. Let
N
γ ( t ) = wOT + ( wCE − wOE )e rt . Then γ ( t O ) = wCT − k , , γ ( t N ) = wCT and
γ ′( t ) = r( γ ( t ) − wOT ). Finally,
wCT
∫
wCT − k
DT ( p )
dp =
p( t ) − wOT
γ ( tN )
∫
γ ( t0 )
t
N
DT ( p )
D ( γ ( t ))
dp = ∫ T
γ ′( t )dt
p − wOT
γ ( t ) − wOT
t0
42
tN
tN
∞
t0
t0
t0
r ∫ DT ( γ ( t ))dt ≥ r ∫ DT ( pOT ( t ))dt ≥ r ∫ d OT ( t )dt = rQ0 ( t 0 ) .■
43
Appendix 2
We show that Corollary 1 of Benveniste and Scheinkman (1979) holds for the problem defined in
equations (1-3). Our maximization problem in (1-3) can be rewritten in their notation as:
Given a technology set T ⊆ ℜ 2 m , find an absolutely continuous path (Qi(t)) which solves
∞
Max
∫ u( Q ( t ),Q& ( t ),t )dt
i
i
0
subject to ( Qi ( t ),Q& i ( t ),t ) ∈ T∀t and Qi(t0) fixed, where Qi=(Q1,..,Qm) is the vector of resource
stocks and Q& i ( t ) = (
n
n
j =1
j =1
∑ q1 j ,..,∑ q mj ). We check that the following conditions hold:
Assumption 1: The pair (Q (t ), Q& (t )) ∈ T ⊆ ℜ + 2 m is convex since resource stocks are non0
negative and finite and T is not empty since there must be a resource with a positive stock.
Assumption 2: The mapping u : T × R → R given by
∑ qij ( t )
u( ⋅,⋅,⋅ ) = e − rt [ ∑
j∈U
i∈ R
0
−1
∫ D j ( x )dx − ∑ ∑ wij qij ( t )] is continuously differentiable on T× R
0
i∈R j∈U
since the demand function is continuous, and u( ⋅,⋅,t ) : T → ℜ is concave since the right hand
side of the above equation is concave (see Theorem 10.7, Rockafellar, 1970).
Assumption 3: An optimal solution {y( t ,Qi ( t 0 )}t =t0 exists and V(Qi) is well-defined in some
∞
neighborhood of Qi(t0) (see Proposition 1).
44
0
Assumption 4. ( Q( t 0 ), y& ( t 0 ,Qi ( t 0 )) ∈ T and the optimal control is a piecewise function of time
(see Seierstad and Sydsaeter, 1987, Theorem 1, p. 75 and Theorem 12, p. 234). ■
45
References
Amigues, J.P., Favard, P., Gaudet, G., Moreaux, M., 1998. On the Optimal Order of Natural
Resource Use When the Capacity of the Inexhaustible Substitute is Constrained, Journal of
Economic Theory 80, 153-70.
Benveniste, L.M., Scheinkman, J.A., 1979. On the Differentiability of the Value Function in
Dynamic Models of Economics, Econometrica 47(3), 727-32.
Benveniste, L.M., Scheinkman, J.A., 1982. Duality Theory for Dynamic Optimization Models of
Economics: The Continuous Time Case, Journal of Economic Theory 27, 1-19.
Bhagwati, J.N., Panagariya, A., Srinivasan, T.N., 1998. Lectures on International Trade, second
edition, MIT Press, Cambridge.
Chakravorty, U., Krulce, D.L., 1994. Heterogenous Demand and Order of Resource Extraction,
Econometrica 62(6), 1445-52.
Chakravorty, U., Roumasset, J.A., Tse, K-P., 1997. Endogenous Substitution among Energy
Resources and Global Warming, Journal of Political Economy 105, 1201-34.
Epstein, L. G., 1978. The Le Chatelier Principle in Optimal Control Problems, Journal of
Economic Theory 19, 103-22.
Farzin, Y.H., 1992. The Time Path of Scarcity Rent in the Theory of Exhaustible Resources,
Economic Journal 102, 813-30.
Gaudet, G., Moreaux, M., Salant, S.W., 2001. Intertemporal Depletion of Resource Sites by
Spatially Distributed Users, American Economic Review 91, 1149-59.
Halvorsen, R., Smith, T.R., 1991. A Test of the Theory of Exhaustible Resources, Quarterly
Journal of Economics 106(1), 123-40.
46
Harberger, A., 1993. The Search for Relevance in Economics, American Economic Review
83(2), 1-16.
Heal, G., 1976. The Relationship between Price and Extraction Cost for a Resource with a
Backstop Technology, Bell Journal of Economics 7(2), 371-78.
Herfindahl, O.C., 1967. Depletion and Economic Theory, in: Gaffney, M., (Ed.), Extractive
Resources and Taxation, University of Wisconsin Press, pp. 63-90.
Hotelling, H., 1931. The Economics of Exhaustible Resources, Journal of Political Economy
39(2), 137-75.
International Energy Agency, 2000. World Energy Outlook: Highlights, Paris.
International Energy Agency, 2002. Key World Energy Statistics, Paris.
Kemp, M.C., Long, N.V., 1980. On Two Folk Theorems Concerning the Extraction of
Exhaustible Resources, Econometrica 48, 663-73.
Lewis, T. R., 1982. Sufficient Conditions for Extracting Least Cost Resource First, Econometrica
50, 1081-83.
Nordhaus, W.D., 1973. The Allocation of Energy Resources, Brookings Papers on Economic
Activity 3, 529-70.
Nordhaus, W. D., 1979. The Efficient Use of Energy Resources, Yale University Press, New
Haven, Conn.
Ricardo, D, 1912. The Principles of Political Economy and Taxation, J.M. Dent & Sons, London.
Rockafellar, T., 1970. Convex Analysis, Princeton University Press, New Jersey.
Rhodes, R., 2002. Energy Transitions: A History Lesson, Keynote address, 6th International
Symposium on Fusion Nuclear Technology, April 8, San Diego.
47
Seierstad, A., Sydsaeter, K., 1987. Optimal Control Theory with Economic Applications, NorthHolland, Amsterdam.
48
Table 1. World Total Final Energy Consumption by Sector (in MTOE): 2000
Sectors
Coal
Crude Oil
Natural
Nuclear,
& Products
Gas
Hydro &
Total
Percent
Share
Others*
Industry
411.5
592.4
491.1
691.1
2186.1
31.6
Transportation 5.9
1701.4
53.7
27.8
1788.8
25.9
Residential &
128.9
655.9
570.3
1575.3
2930.4
42.5
TFC
546.3
2949.7
1115.1
2294.2
6905.3
100
Percentage
7.9
42.7
16.2
33.2
100
Others
Share
*Includes combustible renewables, industrial waste, solar, wind, etc.
Glossary: MTOE: Million Tons of Oil Equivalent; TFC: Total Final Consumption
Note: Electricity consumption is not reported separately since it is an intermediate product. In 2000,
1322.6 MTOE of electricity was produced out of which 42.2% was used in industry and 56% in the
“Residential and Others” sector. Electricity was mainly produced from coal (39%), nuclear, natural gas and
hydro (approximately 17% each) and oil (8%).
Source: Adapted from International Energy Agency (2002).
49
Table 2. Solutions to the two-resource two-demand case
E
T
O
O
C
O
C
C
(a): Oil has dominance and comparative advantage in transportation.
E
T
O
O
O
C
C
C
(b): Oil has dominance and comparative advantage in electricity.
E
T
C
O
O
O
(c): Oil has absolute advantage in transportation and coal in electricity; oil abundant.
E
T
C
O
C
C
(d): Oil has absolute advantage in transportation and coal in electricity; coal abundant.
Notes: E: Electricity; T: Transportation; O: Oil; C: Coal
Another set of four solutions can be generated by switching oil and coal.
50
$
pOE
pOT
pCE
pCT
time
tr
ts
Fig.1. Extraction profile when oil is dominant and has comparative advantage in
transportation. Oil is used exclusively at the beginning and coal at the end.
51
$
φΕ(t)
φΤ(t)
time
t1
t2
k
Fig.2. Oil is dominant and has comparative advantage in transportation (k>0). Oil is
used in electricity until t1 and in transportation until t2.
52
$
pT
pE
pE
pT
time
Fig. 3. Clean oil, dirty coal: the price path for transportation cuts the price of electricity
from below.
53